Abstract
In this paper we present and prove a new alternative weight characterization for the Hardy inequality for decreasing functions. We also give an alternative approach to the characterization of the Hardy inequality using a fairly new equivalence theorem. In fact, this result shows that there are infinitely many possibilities to characterize the considered Hardy inequality for decreasing functions. We also state the corresponding weight characterization for the Pólya-Knopp inequality for decreasing functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Barza, On Some Integral Inequalities Weighted Multidimensional Integral Inequalities and Applications, Ph.D. Thesis, Department of Mathematics, Luleå University of Technology, 199).
A. Gogatishvili, A. Kufner, L.-E. Persson and A. Wedestig, On Some Integral Inequalities An equivalence theorem for integral conditions related to Hardy’s inequality, Real Anal. Exchange 29(2) (2003/04), 867–880.
M. Johansson, Carleman type inequalities and Hardy type inequalities for monotone functions, Ph.D. Thesis, Department of Mathematics, Luleå University of Technology, 2007.
A. Kufner, L.-E. Persson and A. Wedestig, A study of some constants characterizing the weighted Hardy inequality, Orlicz centenary volume, Banach Center Publ, Polish Acad. Sci., Warsaw 64 (2004) 135–146.
A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co, Singapore/ New Jersey / London / Hong Kong, 2003.
L.-E. Persson and V.D. Stepanov, Weighted integral inequalities with the geometric mean operator, J. Inequal. Appl. 7(5) (2002), 727–746.
E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96(2) (1990), 145–158.
V.D. Stepanov, Integral operators on the cone of monotone functions, J. London Math. Soc. 48(2) (1993), 465–487.
V.D. Stepanov, The weighted Hardy’s inequality for nonincreasing functions, Trans. Amer. Math. Soc. 338(1) (1993), 173–186.
A. Wedestig, Some new Hardy type inequalities and their limiting inequalities, J. Inequal. Pure Appl. Math. 4(3) (2003), Article 61.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Johansson, M. (2008). A New Characterization of the Hardy and Its Limit Pólya-Knopp Inequality for Decreasing Functions. In: Bandle, C., Losonczi, L., Gilányi, A., Páles, Z., Plum, M. (eds) Inequalities and Applications. International Series of Numerical Mathematics, vol 157. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8773-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8773-0_10
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-8772-3
Online ISBN: 978-3-7643-8773-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)